Properties

Label 2-1340-1340.959-c0-0-0
Degree $2$
Conductor $1340$
Sign $0.367 - 0.929i$
Analytic cond. $0.668747$
Root an. cond. $0.817769$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.981 + 0.189i)2-s + (−0.771 + 1.68i)3-s + (0.928 − 0.371i)4-s + (−0.959 − 0.281i)5-s + (0.437 − 1.80i)6-s + (0.379 − 1.09i)7-s + (−0.841 + 0.540i)8-s + (−1.60 − 1.84i)9-s + (0.995 + 0.0950i)10-s + (−0.0883 + 1.85i)12-s + (−0.165 + 1.14i)14-s + (1.21 − 1.40i)15-s + (0.723 − 0.690i)16-s + (1.92 + 1.51i)18-s + (−0.995 + 0.0950i)20-s + (1.55 + 1.48i)21-s + ⋯
L(s)  = 1  + (−0.981 + 0.189i)2-s + (−0.771 + 1.68i)3-s + (0.928 − 0.371i)4-s + (−0.959 − 0.281i)5-s + (0.437 − 1.80i)6-s + (0.379 − 1.09i)7-s + (−0.841 + 0.540i)8-s + (−1.60 − 1.84i)9-s + (0.995 + 0.0950i)10-s + (−0.0883 + 1.85i)12-s + (−0.165 + 1.14i)14-s + (1.21 − 1.40i)15-s + (0.723 − 0.690i)16-s + (1.92 + 1.51i)18-s + (−0.995 + 0.0950i)20-s + (1.55 + 1.48i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $0.367 - 0.929i$
Analytic conductor: \(0.668747\)
Root analytic conductor: \(0.817769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1340,\ (\ :0),\ 0.367 - 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4344049863\)
\(L(\frac12)\) \(\approx\) \(0.4344049863\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.981 - 0.189i)T \)
5 \( 1 + (0.959 + 0.281i)T \)
67 \( 1 + (-0.786 - 0.618i)T \)
good3 \( 1 + (0.771 - 1.68i)T + (-0.654 - 0.755i)T^{2} \)
7 \( 1 + (-0.379 + 1.09i)T + (-0.786 - 0.618i)T^{2} \)
11 \( 1 + (0.888 - 0.458i)T^{2} \)
13 \( 1 + (-0.580 + 0.814i)T^{2} \)
17 \( 1 + (-0.723 - 0.690i)T^{2} \)
19 \( 1 + (0.786 - 0.618i)T^{2} \)
23 \( 1 + (-0.580 - 0.814i)T + (-0.327 + 0.945i)T^{2} \)
29 \( 1 + (-0.995 - 1.72i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.580 - 0.814i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-1.39 + 1.09i)T + (0.235 - 0.971i)T^{2} \)
43 \( 1 + (0.223 + 1.55i)T + (-0.959 + 0.281i)T^{2} \)
47 \( 1 + (-0.0947 + 0.00904i)T + (0.981 - 0.189i)T^{2} \)
53 \( 1 + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (-0.415 + 0.909i)T^{2} \)
61 \( 1 + (0.452 - 1.86i)T + (-0.888 - 0.458i)T^{2} \)
71 \( 1 + (-0.723 + 0.690i)T^{2} \)
73 \( 1 + (0.888 + 0.458i)T^{2} \)
79 \( 1 + (0.995 + 0.0950i)T^{2} \)
83 \( 1 + (-0.473 + 0.451i)T + (0.0475 - 0.998i)T^{2} \)
89 \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.23585069471541138157486589185, −9.051593645655366901481277184755, −8.738594174910514193043154166726, −7.53791659365723103115176978061, −6.92248922927466916270662052797, −5.66389156457741647973556787890, −4.90073530724655901632691068775, −4.03409606633960514527375031578, −3.21747751861042197160520266573, −0.883364233615114878068231268540, 0.827002683833521905841396970369, 2.18224846876660759433143488615, 2.89741850868350455381029758067, 4.74528850658623043076730592395, 6.05712739426309576678008425070, 6.47046049950893906971060391785, 7.37806598526667158180131195604, 8.140390648835372711903642712333, 8.348637387732640246535317750317, 9.551466819672296884283749967614

Graph of the $Z$-function along the critical line